Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [390,10,Mod(1,390)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(390, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("390.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 390.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(200.863976104\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 6099x^{2} - 214400x - 1400100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2}\cdot 5\cdot 7 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 6099x^{2} - 214400x - 1400100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 50\nu^{2} - 2799\nu - 8330 ) / 20 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 50\nu^{2} - 3999\nu - 8330 ) / 20 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} - 110\nu^{2} - 13997\nu - 146950 ) / 20 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 60 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - 14\beta_{2} + 5\beta _1 + 18294 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1500\beta_{3} - 9799\beta_{2} + 6499\beta _1 + 9646800 ) / 60 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
16.0000 | −81.0000 | 256.000 | −625.000 | −1296.00 | −9399.64 | 4096.00 | 6561.00 | −10000.0 | ||||||||||||||||||||||||||||||
| 1.2 | 16.0000 | −81.0000 | 256.000 | −625.000 | −1296.00 | −6174.97 | 4096.00 | 6561.00 | −10000.0 | |||||||||||||||||||||||||||||||
| 1.3 | 16.0000 | −81.0000 | 256.000 | −625.000 | −1296.00 | 2837.10 | 4096.00 | 6561.00 | −10000.0 | |||||||||||||||||||||||||||||||
| 1.4 | 16.0000 | −81.0000 | 256.000 | −625.000 | −1296.00 | 5632.50 | 4096.00 | 6561.00 | −10000.0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(1\) |
| \(13\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 390.10.a.f | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 390.10.a.f | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 7105T_{7}^{3} - 57888306T_{7}^{2} - 242714785040T_{7} + 927517492501600 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(390))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{4} \)
|
| $3$ |
\( (T + 81)^{4} \)
|
| $5$ |
\( (T + 625)^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 927517492501600 \)
|
| $11$ |
\( T^{4} + \cdots - 16\!\cdots\!20 \)
|
| $13$ |
\( (T - 28561)^{4} \)
|
| $17$ |
\( T^{4} + \cdots - 12\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots - 33\!\cdots\!32 \)
|
| $23$ |
\( T^{4} + \cdots + 58\!\cdots\!88 \)
|
| $29$ |
\( T^{4} + \cdots + 16\!\cdots\!84 \)
|
| $31$ |
\( T^{4} + \cdots - 19\!\cdots\!68 \)
|
| $37$ |
\( T^{4} + \cdots - 54\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots - 10\!\cdots\!24 \)
|
| $43$ |
\( T^{4} + \cdots + 16\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots - 91\!\cdots\!40 \)
|
| $53$ |
\( T^{4} + \cdots + 54\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 12\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots - 31\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 63\!\cdots\!16 \)
|
| $71$ |
\( T^{4} + \cdots - 36\!\cdots\!12 \)
|
| $73$ |
\( T^{4} + \cdots - 20\!\cdots\!20 \)
|
| $79$ |
\( T^{4} + \cdots + 27\!\cdots\!04 \)
|
| $83$ |
\( T^{4} + \cdots + 60\!\cdots\!48 \)
|
| $89$ |
\( T^{4} + \cdots - 25\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 28\!\cdots\!04 \)
|
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